C.K. GHOSH E L C I T RTICLE R A E R U T A UR tryst with mathematical operations begins with EATURE Onumerical identities. Perhaps the fi rst exercise in addition F that we do is 1 + 1=2, which is the simplest among the numerical identities. Likewise, the multiplication tables provide us with numerous examples of numerical identities. The same is true of subtraction or division, which is repeated subtraction. If we probe into many such identities, we fi nd hidden Pythagoras treasures. The focus of this article is such a treasure hunt. Let us look at the identity, 8 + 16 = 24. It may not reveal any meaningful relation. But instead if we have, Take a look at some numerical identities, 9 + 16 = 25, it becomes quite signifi cant. This is because it can be which with some probing reveal hidden expressed as, treasures. Readers may dig out many more 32 + 42 = 52. such identities. In other words, 3, 4, 5 form a Pythagorean Trio. It indicates that if we have a right-angled triangle with its mutually perpendicular sides of length 3 and 4 units, then its hypotenuse The right hand side of identity (a) is 273, which is a number is bound to be of length 5 units. exactly divisible by ‘7’. The le hand side is the sum of the Starting from (3, 4, 5) we may generate many more such number of days in January, February, March, April, May, June, trios, by multiplying each member by 2, 3, 4 and so on. The trios July, August, September of any ordinary year (which is not a leap thus obtained would be (6, 8, 10); (9, 12, 15); (12, 16, 20) and year). So for every such year the sum will always be 273 (= 7 × so on. These are Non-Primitive Pythagorean Trios. There can be 39), which means that the total number of days from 1st January many Primitive Pythagorean Trios, such as (5, 12, 13); (7, 24, 25); to 30th September, both inclusive, consists of exactly 39 weeks. (8, 15, 17); (9, 40, 41) and so on. It can be seen that, So, the dates 1st January and the one following 30th September, i.e. 1st October will always fall on the same day. So, the calendar 25 + 144 = 169, i.e. 52 + 122 = 132 of January and October would always be the same for a normal 49 + 576 = 625, i.e. 72 + 242 = 252 year. It may be verifi ed from any calendar. 64 + 225 = 289, i.e. 82 + 152 = 172 The identity (b) indicates that the sum of the number of 81 + 1600 = 1681, i.e. 92 + 402 = 412 days of April, May, June or September, October, November is always ‘91’ (7 × 13), that is, there are exactly 13 weeks between 1st So, the identities mentioned above, would appear to be quite April and 30th June and so also between 1st September and 30th insignifi cant, but there is a hidden treasure in each one of them November, thereby indicating that the calendars of the months which speaks of the phenomenal contribution to mathematics April, July and September, December are always the same. made by Pythagoras. On similar lines, we may examine the following identities: Through his famous theorem, Pythagoras established (c) 31 + 28 + 31 + 30 = 120 = 119 + 1 = 17 × 7 + 1 a vital link between geometry and numbers. Apparently the (d) 31 + 29 + 31 + 30 = 121 = 119 + 2 = 17 × 7 + 2 domains of geometry and numbers seem to be de-linked; but They indicate that the number of days from 1st January to we fi nd that there indeed exists a connection, the testimony to 30th April of a normal year is just ‘1’ more than a multiple of ‘7’ which was provided by the numerical identities involving the and in a leap year it is ‘2’ more than a multiple of ‘7’. So, the day Pythagorean Trios. following 30th April, that is, 1st May will be one-day or two-day Let us now examine the two identities given in the box ahead of 1st January depending on whether it is a normal year below: or a leap year. (a) 31 + 28 + 31 + 30 + 31 + 30 + 31 + 31 + 30 = 273 Now, the number of days in a normal year is 365 (= 52 × 7 + (b) 30 + 31 + 30 = 91 1) and that in a leap year is 366 (= 52 × 7 + 2), which indicates that Science Reporter, OCTOBER 2016 44 FEATURE ARTICLE 1st January of the coming year will be one day ahead of that of the present year if it is a normal year and two days ahead if it is a leap year. Therefore, the calendar of the month of May of the present year is always the same as that of January of the coming year irrespective of it being a normal or a leap year. Many readers might have had the experience of not receiving calendars towards the beginning of a year. Notwithstanding the fact that these days calendars can be obtained just by the touch of a fi nger on a smart-phone, the above phenomenon, based on simple numerical identities stated above, turns out to be very handy in handling problems arising out of absence of calendars during the initial days of the year. ‘May’ of the current year Johann Jakob Balmer Johannes Rydberg becomes ‘January’ of the next year, naturally ‘June’ of the current year becomes ‘February’ of the next year. Likewise, many other similarities can be obtained. Narcissism Through Identities of diff erent nations stipulates that the diff erence of time between two countries should essentially be a multiple of 30 minutes. Narcissism is the pursuit of gratifi cation from vanity or egotistic For example, Singapore and Bangladesh are respectively 150 admiration of one’s own a ributes. The origin of the term lies in minutes and 30 minutes ahead of us, whereas UK and Muscat are Greek mythology, where the young Narcissus fell in love with behind us by 270 minutes and 90 minutes respectively. The Earth his own image refl ected in a pool of water. rotates through 360 degrees in 24 hours, i.e. 1440 minutes. So, a Narcissism is exhibited by several numbers. Let us look at diff erence of one degree in longitude is equivalent to a separation the following identities: of 1440 ÷ 360 = four minutes. So, 30 minutes is equivalent to 30 (e) 1 + 125 + 27 = 153 ÷ 4 = 7.5 degrees. Thus, it turns out that the diff erence of time (f) 27 + 343 + 0 = 370 between any two countries should be a multiple of 7.5 degrees. (g) 27 + 343 + 1 = 371 Now, the easternmost point of India is just east of Kibithu, (h) 64 + 0 + 343 = 407 Arunachal Pradesh, longitude – 96.5 Degrees (approx) East, and the westernmost point is just West of Ghuar Mota, Gujarat, (i) 1 + 1296 + 81 + 256 = 1634 longitude – 68.5 Degrees (approx) East. Thus, India stretches (j) 4096 + 16 + 0 + 4096 = 8208 between the longitudes 68.5 degrees (E) to 96.5 Degrees (E) from (k) 6561 + 256 + 2401 + 256 = 9474 West to East. The available multiples of 7.5 between 68.5 and 96.5 (l) 93 = 729 in ascending order are 75(10 × 7.5); 82.5(11 × 7.5) and 90(12 × 0 0 0 (m) 83 = 512 7.5). So, the available longitudes are 75 E, 82.5 E and 90 E. 9 × 7.5 (= 67.5) and 13 × 7.5 = (97.5) just miss the country boundary (n) 9 × 16 = 144 respectively on the western and the eastern sides. Each of the ten (e) to (n) above identities appear like any The central among these is 82.50E, which happens to be other of its kind. But a careful observation would reveal that they approximately the longitude of Mirzapur (the actual longitude refl ect narcissism. We can see that if we rewrite them as under: of Mirzapur is 82.580E). Since independence in 1947, a clock 3 3 3 (e) 1 + 5 + 3 = 153 tower in the Shankargarh Fort in Mirzapur in Allahabad district (f) 33 + 73 + 03 = 370 of U ar Pradesh has been being considered as the reference for (g) 33 + 73 + 13 = 371 Indian Standard Time. It shows that we are 11 × 30 = 330 minutes (h) 43 + 03 + 73 = 407 or fi ve and half hours ahead of Greenwich. Thus, the identity (o) is instrumental behind locating Mirzapur as the reference for (i) 14 + 64 + 34 + 44 = 1634 IST. (j) 84 + 24 + 04 + 84 = 8208 (k) 94 + 44 + 74 + 44 = 9474 Identities Reveal Physical Phenomenon (l) (7 + 2)√9 = 729 Let us now concentrate on some identities involving inverse (5+1)/2 (m) (5 + 1 + 2) = 512 squares of integers as given below: (n) (1 + 4 + 4) × (1 × 4 × 4) = 144 Now, let us examine an identity in the form of an innocuous 1 1 1 1 5 (p) 2 - 2 = - = product given in the box below: 2 3 4 9 36 1 1 1 1 3 (o) 11 × 7.5 = 82.5 (q) - = - = 22 42 4 16 16 One may wonder what it signifi es. It is instrumental in 1 1 1 1 21 identifying the town of Mirzapur (near Allahabad) as the (r) 2 - 2 = - = reference for determining the Indian Standard Time (IST). 2 5 4 25 100 Let us see how. 1 1 1 1 2 (s) 2 - 2 = - = The international law regarding fi xation of standard times 2 6 4 36 9 45 Science Reporter, OCTOBER 2016 FEATURE ARTICLE to be the Fundamental Rydberg Frequency for the K-alpha lines.